A222591 Numerators of (n*(n - 3)/6) + 1, arising as the maximum possible number of triple lines for an n-element set.

1, 5, 8, 4, 17, 23, 10, 38, 47, 19, 68, 80, 31, 107, 122, 46, 155, 173, 64, 212, 233, 85, 278, 302, 109, 353, 380, 136, 437, 467, 166, 530, 563, 199, 632, 668, 235, 743, 782, 274, 863, 905, 316, 992, 1037, 361, 1130, 1178, 409, 1277, 1328

Offset: 3

Author

Jonathan Vos Post, Feb 25 2013

Keywords

• nonn
• easy
• frac

Comments

Numerators of (n*(n - 3)/6) + 1, which arises as the maximum possible number of triple lines for an n-element set, according to Green and Tao, cited in Elekes. The fractions for n = 3, 4, 5, 6, ... are 1/1, 5/3, 8/3, 4/1, 17/3, 23/3, 10/1, 38/3, 47/3, 19/1, 68/3, 80/3, 31/1, 107/3, 122/3, 46/1, 155/3, 173/3, 64/1, 212/3, 233/3, 85/1, 278/3, 302/3, 109/1, 353/3, 380/3, 136/1, 437/3, 467/3, 166/1, 530/3, 563/3, 199/1, 632/3, 668/3, 235/1, 743/3, 782/3, 274/1, 863/3, 905/3, 316/1, 992/3, 1037/3, 361/1, 1130/3, 1178/3, 409/1, 1277/3, 1328/3. The corresponding denominators are A169609.

Examples

			a(10) = 38 because (10*(10 - 3)/6) + 1 = 38/3.
		

Links

• György Elekes, Endre Szabó, On Triple Lines and Cubic Curves --- the Orchard Problem revisited, arXiv:1302.5777 [math.CO], Feb 23, 2013.
• Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Crossrefs

Cf. A169609.

Programs

Mathematica

Numerator[Table[(n(n-3))/6+1,{n,3,60}]] (* or *) LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{1,5,8,4,17,23,10,38,47},60] (* Harvey P. Dale, Feb 11 2015 *)